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Full text of "Treatise On Analysis Vol-Ii"

300       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

(a)   Let/be a continuous function on G, with values in [0, 1] and support contained
in V, and such that/(*?)> 0. Put

Show that for each z e G we have ||Y(Z)#  g\\ <; c\\y(z)/-/||, where c = \f(s) df$(s),

and that g(z) < g(e) if z ^ e. (If not, we should have y(z~1)f=f, which would imply
that z* e V for all k.)

(b) Let (fn) be a uniformly bounded equicontinuous family of positive functions
with support contained in V, and suppose that for each n there exists a subset An <= V
such that ||Y(*)/ ~/J ^ M for all n and all z E An. Show that the set of functions

is equicontinuous, and that the same is true of the set of functions j(z)gn  gn where
z e An , for each n.

10.   Let G be a locally compact group and let ut v be two positive bounded functions on G.
For each x e G let

w(x) = sup u(y)v(y'lx).

yeG

(a)   Show that w(x) = sup u(xy~l)v(y), and that for each s e G we have

yeG

If A, B are the supports of u, v respectively, show that the support of w is contained

in the closure of AB.

(b)   Suppose that v(x~ l) = v(x). Show that, for all x, x' in G, we have

MX)- w(x')\ ^||
(Observe that for each y e G we have

Y(*'MI  u(y)(v(y~lx) -

In particular, if v is continuous and compactly supported, then so is w.

11. Let G be a locally compact group, and V a symmetric compact neighborhood of e in
G such that V2 contains no subgroup ={e}. For each integer /, let U/ be the set of
points x such that xk e V for 1 ^ k < /. As / - + o, the sets U/ form a fundamental
system of neighborhoods of e in G (Section 12.9, Problem 6b)). Let nt be the largest
integer n such that U" <= V. Define a function wf on G as follows :

(a)   Show that for all ;*:  U( we haveinto m parts by means of an increasing sequence